2008/42
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How do epidemics induce behavioral changes?
Raouf Boucekkine, Rodolphe DesbordesCORE
Voie du Roman Pays 34
B-1348 Louvain-la-Neuve, Belgium. Tel (32 10) 47 43 04
Fax (32 10) 47 43 01
E-mail: [email protected] http://www.uclouvain.be/en-44508.html
CORE DISCUSSION PAPER 2008/42
How do epidemics induce behavioral changes?
Raouf BOUCEKKINE1, Rodolphe DESBORDES2
and Hélène LATZER 3
July 2008
Abstract
This paper develops a theory of optimal fertility behavior under mortality shocks. In a 3-periods OLG model, young adults determine their optimal fertility, labor supply and life-cycle consumption with both exogenous child and adult mortality risks. For fixed prices (real wages and interest rate), it is shown that both child and adult one-period mortality shocks raise fertility due to insurance and life-cycle mechanisms respectively. In general equilibrium, adult mortality shocks give rise to price effects (notably through rising wages) lowering fertility, in contrast to child mortality shocks. We complement our theory with an empirical analysis on a sample of 39 Sub-Saharan African countries over the 1980-2004 period, checking for the overall effects of the adult and child mortality channels on optimal fertility behavior. We find child mortality to exert a robust, positive impact on fertility, whereas the reverse is true for adult mortality. We further find this negative effect on fertility of a rise in adult mortality to
dominate in the long-term the positive effect on demand for children resulting from an
increase in child mortality.
Keywords: fertility, mortality, epidemics, HIV JEL Classification: J13, J22, O41
1 CORE and IRES, Université catholique de Louvain, Belgium and University of Glasgow. E-mail:
[email protected]. This author is also member of ECORE, the newly created association between CORE and ECARES.
2 University of Strathclyde, U.K. E-mail: [email protected]
3 IRES, Université catholique de Louvain, Belgium. E-mail: [email protected]
We are grateful to two anonymous referees for numerous suggestions which have decisively shaped this version. We acknowledge the financial support of the Belgian research programmes PAI P5/10 and ARC 03/08-302. The usual disclaimer applies.
This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.
1 Introduction
Recently, a controversy has been taking place around the fertility impact of the HIV/AIDS epidemic. While the study of Young (2005) concerning the South-African case has concluded that the epidemic leads to a decrease in fertility, Kalemli-Ozcan (2008a) has identi ed the opposite eect on a panel of African countries over the 1985-2004 period. At the heart of both Young (2005) and Kalemli-Ozcan (2008a)'s arguments lies the existence of behavioral responses to the epidemics going beyond the simple sexual behavior adjustment to this sexually-transmitted
disease. 1 In other words, both authors claim households will purposefully adapt their fertility
choices. Indeed, Kalemli-Ozcan (2008a)'s empirical analysis concludes in favor of a positive eect on fertility of HIV/AIDS despite the negative impacts of the biological and sexual responses to
the epidemics,2 while Young (2005), using microdata of Sub-Saharan African countries, nds
that higher HIV prevalence leads to an increased use of all forms of contraception, con rming that the behavioral changes induced by the epidemics go beyond a simple willingness to reduce unprotected sexual activity.
Young (2005) explains this change in individual behaviors by a rise in the opportunity cost of rearing children, driving down the fertility decisions of women. At the heart of his argument lies the idea that, similarly to what occurred during the Black Death in the fourteenth century, the heavy labor shortage induced by epidemics of the type of HIV/AIDS essentially killing active adults should have a signi cant positive eect on wages, which might well raise female participation in the labor market and ultimately lead to a lower fertility. Set in more theoretical terms in line with Kremer and Chen (2002), it might be the case that the substitution eect induced by higher wages (lowering fertility through the rising opportunity cost of rearing children) ends up dominating the income eect (increasing fertility through a higher amount of resources available to bring up children). On the other hand, Kalemli-Ozcan (2008a) relates the fertility
response to both a shock on adult longevity 3 and to the survival probability of the children.
As far as child mortality is concerned, her argument is tightly linked to the insurance eect (Kalemli-Ozcan, 2002), deriving from an increasing uncertainty about children's survival to the adult age: the perception of a rise in child mortality may hence generate a further rationale for increasing the number of osprings. The ongoing AIDS-fertility debate hence seems to oppose two eects at work in the case of an epidemic shock: on one side the direct, positive impacts on fertility described by Kalemli-Ozcan (2008a) of a diminishing adult survival rate (adult mortality shock) and a decreasing survival probability of children (child mortality shock), and on the other side the indirect, negative wage eect on fertility put forward by Young (2005), sort of second-round response to the initial adult mortality shock.
1Kalemli-Ozcan (2008a) provides an overview of the medical literature and concludes that uninfected people,
or people believing not to be infected, do not modify their fertility-related behaviors: indeed, in the African case, numerous social and political circumstances are particularly unfavorable to awareness of HIV/AIDS (World Bank, 1997). In particular, Oster (2005), using DHS data on sexual behavior from a subset of African countries, nds that sexual behavior has only been slightly modi ed since the appearance of the epidemic.
2As far as the biological changes are concerned, the fertility of HIV-positive women has been found to be
about 30% lower than HIV-negative women in SSA, causing a population-attributable decline in total fertility of 0.37% for every percentage point of HIV prevalence (Lewis et al., 2004). This negative impact on fertility is usually explained by lower coital frequency due to ill health and epidemiological synergies between HIV and other sexually-transmitted infections, which reduce the ability to conceive and increase the risk of foetal loss (Gray et al., 1998).
3However, even though she also acknowledges the role of adult mortality shocks as in Young (2005), she argues
against the latter that they have a positive eect on fertility decisions, resulting from a quantity/quality trade-o operated by adults once their life span is reduced: they end up investing less in their own education and having more children (Soares, 2005).
We argue along with Young (2005) that the fertility impact of epidemics cannot be nely appre-hended by only considering decreasing survival rates, whether it be for adults or children: as far as one is concerned with fertility choices under epidemics, the price eects (and essentially the wage eect) should be accurately isolated. The aim of this paper is to capture those two dierent types of eects on fertility response to an epidemic shock (the direct mortality eects and the indirect price eects) in a comprehensive, intertemporal general equilibrium framework that will allow us to both precisely decompose the dierent channels at work and analyze the existing interactions between them. In other words, we aim at studying the in uence of mortality shocks in a general equilibrium framework allowing for price eects to arise.
We shall proceed in two steps. We rst present an overlapping generations model depicting the individual behavior modi cations in response to an epidemic shock. In order to clearly identify the dierent channels at work in the case of an epidemic shock and their consequences on fertility behavior, we proceed progressively. We rst place ourselves in a partial equilibrium framework, where prices (basically the real wage and the interest rate) are given. In this partial equilibrium analysis, we rst characterize the optimal fertility, labor supply and saving under transitory mortality shocks. It is of utmost importance to notice that our model encompasses and clearly dierentiates the two possible mortality channels in the case of an epidemic such as the one of HIV/AIDS, i.e. rst a decrease in the life span (adult mortality shock) and second a lower probability of child survival (child mortality shock). We show that the two types of mortality shocks induce an increase in fertility and a lower labor supply, but that the underlying mechanisms are radically dierent: while the lower survival probability of children triggers an insurance eect à la Kalemli-Ozcan (2008b), the shock on adult mortality induces a life-cycle eect where the reduction in chances to reach senior age decreases the need for life-cycle consumption and increases the desired number of osprings. We then study how the optimal fertility and labor supply behaviors are modi ed when we allow for the prices (wages and interest rates) to exogenously move: this part can be considered as an intermediary step towards our nal, general equilibrium analysis where those prices will endogenously adapt. We particularly show that under certain parametric conditions, the increase in wages resulting from labor shortage can end up reducing fertility and raising labor supply. Hence, we show that a consequence from the initial adult mortality shock (i.e. the resulting upward adjustment in wages) might end up having opposite eects on the fertility behavior. We then nally take the analysis a step further by closing the model, studying a general equilibrium version where prices (wages and interest rates) are endogenously determined: we then show that child mortality shocks do not have any impact on prices in general equilibrium, and hence unambiguously lead to a rise in fertility. On the other hand, we exemplify the existence of a price eect in the case of a shock on adult mortality, since the latter is found to cause wages to go up: therefore, our model clearly demonstrates that a rise in adult mortality has an ambiguous eect on fertility in a general equilibrium setting. We deem this opposition between the child and adult mortality general equilibrium eects to be the key result of our theoretical model.
In a second step, acknowledging the di culty to provide a simple and elegant analytical char-acterization of the cases where the wage eect dominates the mortality eects, we complement our theoretical model with an empirical contribution to the AIDS-fertility debate. We conduct an empirical analysis on a sample of 39 SSA countries over the 1980-2004 period, checking for the overall eects of the adult and child mortality channels on optimal fertility behavior. We nd child mortality to exert a robust, positive impact on fertility, whereas the reverse is true for adult mortality. Hence, to interpret those results in the line of our model, it seems that in the case of the HIV/AIDS epidemic in Sub-Saharan countries, the indirect wage eect outweighs the
direct "life-cycle eect" in the case of an adult mortality shock. We further nd this negative eect on fertility of a rise in adult mortality to dominate in the long-term the positive eect on demand for children resulting from an increase in infant mortality. Even though these results tend to support the ndings of Young (2005, 2007), it must nevertheless be emphasized that we also show they do not contradict the argumentation of Kalemli-Ozcan (2008a) concerning the eects of child mortality, since an increase in precautionary demand for children following a shock on child mortality is also exempli ed by our empirical results. However, we nd this eect to be eventually outweighed by the full impact of a rise in HIV prevalence on adult mortality. To summarize, the original contributions of this paper are twofold. We rst provide a compre-hensive intertemporal, general equilibrium framework for the study of epidemics, encompassing not only the direct eects of mortality shocks (both child and adult) but also the second-round eects occurring once you allow for endogenously determined equilibrium prices. We are hence able to exemplify the opposite general equilibrium eects of child and adult mortality shocks: while the rst one is found to exert an unambiguously positive eect on fertility through the absence of price eect, the eect on fertility of the second one is ambiguous. Beyond that novel
contribution to the literature on life expectancy and growth4, we believe this paper then also
sheds light on the particular ongoing debate between Kalemli-Ozcan (2008a) and Young (2005) in the HIV/AIDS case. First, our theoretical part provides a sort of synthesis of the 2 contribu-tions, exemplifying the co-existence of all the eects identi ed by the two papers. Our empirical part then con rms the existence of both an increase in precautionary demand for children fol-lowing a rise in child mortality and a negative in uence on fertility of a rise in adult mortality, which eventually is found to dominate in the long-run.
The paper is organized as follows. Sections 2 and 3 present our theoretical model and its predictions concerning rational behavior under epidemics, rst in a partial equilibrium setting with xed prices (section 2) and then moving to general equilibrium results (section 3). Section 4 presents our empirical exercise and reports the results obtained. Section 5 concludes.
2 Rational behavior under epidemics: partial equilibrium
theory
In order to clearly identify the dierent forces at work in the in uence of an epidemic shock on the fertility and labor supply behavior, we shall start with a partial equilibrium analysis where prices (basically the real wage and interest rate) are given. Within this simple framework, we rst characterize optimal fertility, labor supply and saving behavior under transitory mortality shocks. In a second step, we also study in this partial equilibrium framework how this behavior is altered if additionally the epidemic shocks also exogenously modify prices, especially real wages. We study in particular the in uence of the wage eect, de ned as the increase in real wages induced by the heavy shortage in labor supply resulting from massive epidemics like the Black Death or AIDS. This positive eect on wages is not only emphasized by Young (2005) in his inspection of the South African AIDS tragedy, but is also commonly admitted by sociologists
and historians (Herlihy, 1997) concerning epidemics such as the Black Death.5
4While a comprehensive theoretical analysis of the relationship between life expectancy, education and fertility
has already been provided by Hazan and Zoabi (2006), our model contributes to the literature by tackling the problem of behavioral answer to dierent types of mortality shocks, and studying in a transparent and tractable way the combined impact of a shorter survival probability and a positive eect on wages.
The model presented here can be seen as a reasonable combination of the Young (2005) static model and Zhang and Zhang (2005) overlapping generations model. It is however important to signal right away that in order to keep this analytical part as algebraically transparent as possible, we don't explicitly allow for the traditional trade-o quality vs quantity of children that is present in both mentioned models through the educational choices of the parents. Indeed, for the sake of computational clarity, we here choose to focus on the fertility decision (quantity of children), not including an education variable. One should however notice that adding such a trade-o to our theoretical set-up would deliver the typical outcome that in response to the epidemic shocks, parents move the quality and quantity of children in opposite directions. Hence, implications for optimal education decisions will result quite naturally once we have determined the optimal fertility behavior.
Last but not least, it should be noted that we depart from the previous models by including exogenous child mortality, which is a crucial part of our story. This modeling aspect has been treated in the spirit of Kalemli-Ozcan (2008b), with a child mortality shock occurring at the beginning of the period and hence a rearing eort being made only in the case of surviving children.
2.1 The model
The model is a 3 periods, one good overlapping generations model. An individual born in period
t has a probability qt to survive to the young adult age in period t + 1, and conditionally to
this survival, he can live as a senior adult in period t + 2 with probability pt+1. In the rst
period of life (childhood), the individual spends all his time endowment (say one unit of time)
having leisure. In period t + 1, he becomes a young adult with (exogenous) probability qt. A
surviving young adult consumes ct+1, has nt+1 children, works a proportion lt+1 of his unit time
endowment, paid at an exogenous wage per unit of time wt+1, and saves st+1 for consumption in
his old age (provided he survives to this age). In our simple model, only young adults work, there is no child labor, and we also disregard social security mechanisms. As usual, having children is costly: in our model, raising children costs a certain amount of the overall time endowment. We assume, in the spirit of Kalemli-Ozcan (2008b), that child mortality occurs at the beginning
of the period, and is observed by parents before they decide about their rearing eort.6 The
rearing eort is therefore assumed to depend only on surviving children.7 To simplify, we model
this cost as a proportion θqt+1nt+1 of time endowed, with θ > 0. The young adult then becomes
a senior adult with probability pt+1. A senior adult consumes ct+2 out of the savings made in
t + 1. A classic feature in this kind of framework is then to assume the existence of an annuity market which guarantees that survivors get the savings plus interests of the young adults who
die before reaching the seniority. Accordingly, the return rate to savings is given by Rt+2
pt+1.
As already signaled and justi ed, with respect to Zhang and Zhang (2005), one can notice
6In Kalemli-Ozcan (2008b), it should be noted that both fertility and education decisions are modeled, with
parents operating fertility choices before the uncertainty is realized, as opposed to investing only in the human capital of surviving children (education choice). As it has already been signalled, our model however does not explicitly allow for educational choices, and we hence consider rearing costs in their broadest acception of the overall time parents have to dedicate to a child if this one survives. Hence, we model the rearing eort decision as being similar in its timing to the education eort decision in Kalemli-Ozcan (2008b).
7One can consider the alternative assumption of parents deciding about their rearing eort before the
uncer-tainty is realized. This case is studied in Kalemli-Ozcan (2003), where both the fertility and education optimal choices are made simultaneously under uncertainty about the number of surviving children. In our set-up, it would however change markedly the algebra and complicate the technical discussion without changing the main sensitive point of the paper on the contrasting behavioral impacts of adult Vs child mortality.
that we omit schooling and human capital accumulation. Indeed, introducing them would result in a system of 7 equations as opposed to 5 below: the economic gains from such an extension are completely overshadowed by the algebraic cost. Instead we choose to stick to a simpler model, that will guarantee a precise identi cation of the eects at work concerning the fertility and labor supply decisions under epidemic shocks. A more fundamental dierence that we will comment now comes from preferences, whose speci cation is closer to the spirit of Beckerian behavioral models such as the one developed by Young (2005). Indeed, we assume that the preferences of a (surviving) young adult individual born in t are given by:
U (ct+1, nt+1, lt+1, ct+2) = c1−σc t+1 1 − σc +α1 (qt+1nt+1)1−σn 1 − σn +α2 (1 − θqt+1nt+1− lt+1)1−σl 1 − σl +pt+1 α3 c1−σc t+2 1 − σc
where αi, i = 1, 2, 3, are strictly positive constants, and σc, σn and σl, are the usual positive
elasticity parameters respectively related to consumption (either young or old consumption),
number of children and leisure.8
Hence, consistently with Young's static preferences, and since labor supply response to epi-demic shocks is a fundamental aspect of our paper, we introduce disutility of working (and rearing children) in the second period of life: young adults enjoy leisure, and working and rear-ing children reduces their utility. In contrast, Zhang and Zhang (2005) introduce such a leisure term for children, along with their purpose of featuring educational choices. Also notice that utility depends on the number of surviving children, a feature which can be found either in Kalemli-Ozcan (2008b) or in evolutionary biology-based growth models à la Galor and Moav (2002).
Finally, the general iso-elastic speci cation for utility terms is aimed for generality. As we shall see later, some important behavioral implications of epidemics do depend on the elasticity parameters.
The budgetary constraints for periods t + 1 and t + 2 are as follow:
ct+1+ st+1 = wt+1lt+1, (1)
ct+2 =
Rt+2
pt+1
st+1, (2)
inducing the intertemporal budgetary constraint
ct+1+
pt+1
Rt+2
ct+2 = wt+1lt+1. (3)
Optimal behavior is obtained by maximization of the utility function with respect to the
four decision variables (ct+1, nt+1, lt+1, ct+2)under the constraint (3). The resulting optimization
problem is as follows. Call λt+1 the Lagrange multiplier associated to (3). The rst-order
8Of course, in case one of these elasticity parameters is equal to 1, the corresponding utility term becomes
conditions with respect to the four variables above in this order are: c−σc t+1 = λt+1, (4) θ α2 qt+1σn (1 − θqt+1nt+1− lt+1)−σl = α1 n−σt+1n, (5) α2 (1 − θqt+1nt+1− lt+1)−σl = λt+1wt+1, (6) α3 c−σt+2c = λt+1 Rt+2 . (7)
Characterizing optimal behavior amounts to solving the system (3) to (7) in the ve variables
(ct+1, nt+1, lt+1, ct+2, λt+1), all strictly positive, under the time resource constraint, θ qt+1 nt+1+
lt+1 < 1, for given prices wt+1 and Rt+2, and given probabilities qt+1 and pt+1. The rst
propo-sition shows that this problem has a unique solution.
Proposition 1 The system (3) to (7) has a unique solution in (ct+1, nt+1, lt+1, ct+2, λt+1), all
strictly positive, satisfying θ qt+1 nt+1+ lt+1 < 1.
Proof: Combining (3), (4) and (7), one can nd
ct+1= wt+1lt+1 1 + pt+1α 1 σc 3 R 1 σc−1 t+2 . (8)
Now, combining (5) and (6), one gets
α1 n−σt+1n = θ qσt+1n λt+1 wt+1,
which yields by (4) and (8)
α1 n−σt+1n = θ qσn t+1w1−σct+1 µ 1 + pt+1 α 1 σc 3 R 1 σc−1 t+2 ¶−σc l−σc t+1. (9)
Now, it is straightforward to see that the proposition is done if we prove that the system (5)-(9)
admits a unique solution in nt+1 and lt+1 satisfying θ qt+1 nt+1+ lt+1 < 1. Using (9) to express
lt+1 as a function of nt+1, and substituting this function in (5), we get a single equation in nt+1,
which is fundamental to our purposes:
α1 n−σt+1n = θ α2qσt+1n · 1 − θ qt+1nt+1− q σn σc t+1Ω −1 σc t+2 w 1 σc−1 t+1 n σn σc t+1 ¸−σl , (10) where Ωt+2 = αθ1 µ 1 + pt+1α 1 σc 3 R 1 σc−1 t+2 ¶−σc
. Denote by ¯nt+1, the number of children satisfying
the equality (implying zero leisure): θqt+1n¯t+1+ q
σn σc t+1 Ω −1 σc t+2 w 1 σc−1 t+1 n¯ σn σc t+1 = 1. On the interval
(0; ¯nt+1), the left-hand side of (10) is a strictly decreasing function from in nity to α1 n¯−σt+1n while
the right-hand side is increasing from θ α2 qσt+1n to in nity. Therefore, they should be equal at a
We now study the impact of epidemics on the optimal decisions featured in Proposition 1,
i.e. the eects on individual behavior of a drop in the survival probabilities pt+1 and qt+1 for
given prices. Then we analyze how this optimal behavior is altered when the latter prices also (exogenously) move.
2.2 Optimal behavior under mortality shocks
We consider an epidemic shock aecting only the generation born in period t. More precisely, and consistently with AIDS epidemiology, the epidemic rst hits the young adults of this generation,
lowering the survival probability pt+1, then their children, causing the survival probability qt+1
to drop. Considering longer epidemic episodes makes the general equilibrium study analytically intractable as one can see in the next section. In this section, we also assume that the prices
wt+1 and Rt+2 are xed.
Suppose the survival probability pt+1goes down. Then, optimal fertility as given by equation
(10) is necessarily altered via the term Ωt+2. The same term will also modify labor supply and
then savings, as re ected in the following proposition.
Proposition 2 Under given prices wt+1 and Rt+2, a decrease in the survival probability pt+1
always raises fertility nt+1 and reduces labor supply lt+1 and savings st+1, for any σc positive.
Proof: Let us start with optimal fertility response. Indeed, a change in pt+1 does not aect
the left hand side of (10), it only aects the right-hand side through the term Ωt+2. Because
Ω− 1 σc t+2 = ³α1 θ ´−1 σc · 1 + pt+1 α 1 σc 3 R 1 σc−1 t+2 ¸ ,
it follows that a drop in the survival probability pt+1will decrease the right-hand side of equation
(10). Because the left-hand side is unaected, and the right-hand side is increasing in nt+1, the
equality (10) is re-established if and only if optimal fertility rises.
To get the property relative to labor supply, we can use a similar argument. First obtain the
corresponding single equation in lt+1 combining (5) and (9):
α1 Ωt+2 w1−σct+1 l−σc t+1 = θ α2 · 1 − θΩ 1 σn t+2w σc−1 σn t+1 l σc σn t+1− lt+1 ¸−σl , (11)
Then apply the same kind of reasoning as just above on (10). Focusing on equation (11), one
can see that the left-hand side is shifted downwards when pt+1 drops. Since the function in the
left-hand side is decreasing in lt+1, this means that lt+1 has to decrease to re-establish equation
(11) for an unchanged right-hand side. However, in contrast to the fertility analysis above, both
sides of the equation (11) are altered. Indeed, a drop in pt+1 also causes the right-hand side
to shift upwards, via the term Ωσn1
t+2, inducing an additional downward move in lt+1 since the
right-hand side is increasing in this variable.
It remains to depict how savings are altered. Using (1) and (8), one can express savings as:
st+1= pt+1α 1 σc 3 R 1 σc−1 t+2 1 + pt+1α 1 σc 3 R 1 σc−1 t+2 wt+1lt+1. (12)
It follows that savings drop when the survival probability goes down for two reasons. On one hand, as demonstrated above, labor supply diminishes following a drop in the adult survival
probability, triggering a decrease in the overall available resources. On the other hand, pt+1 has
a direct eect on savings via the term pt+1α
1 σc 3 R 1 σc −1 t+2 1+pt+1α 1 σc 3 R 1 σc −1 t+2
, which is an increasing function of pt+1
for a given interest rate. This direct eect simply features that since young adults only save to consume in their senior age, a diminishing probability of survival should lower the incentives to save. Hence, savings unambiguously decrease. ¤
Our results complement those of Zhang and Zhang (2005) to a certain extent.9 Indeed, even
though their model is built in a general equilibrium framework, preferences are logarithmic, which has the consequence of neutralizing the general equilibrium eects that might arise through an impact on equilibrium prices, as we shall neatly show in the next section. In such a case, and in ours since we focus on partial equilibrium results in this section, prices are neutralized, and so we can compare the mechanisms at work. In both papers, the reduction in survival probability diminishes the need for life-cycle consumption with respect to fertility, which in Zhang and Zhang (2005) implies an increase in fertility, a decrease in savings and a drop in labor supply in the absence of disutility of working and rearing children. In our model, the disutility of working and rearing children is present, and the properties still hold, which ultimately shows their robustness.
More importantly, our results point out that fertility does increase under adult mortality shocks just like it has been shown to behave under child mortality shocks by Kalemli-Ozcan (2002, 2008a,b). In our case, it happens because of a clearly identi ed life-cycle eect, acting through a diminishing need for life-cycle consumption, while an insurance eect seems to play in the case of child mortality. Since we have also incorporated child mortality à la Kalemli-Ozcan (2008b), we shall go a step further and provide a more comprehensive picture of the impact of child mortality on optimal behavior in our model. The results are re ected in the following proposition.
Proposition 3 Under given prices wt+1 and Rt+2, a decrease in the survival probability qt+1
always raises fertility nt+1 but leaves labor supply lt+1 and savings st+1 unchanged for any σc
positive. Moreover, a decrease in qt+1 is exactly compensated by a proportional rise in fertility.
Proof: The proof is very simple. Recall equation (10):
α1 n−σt+1n = θ α2qσt+1n · 1 − θ qt+1nt+1− q σn σc t+1Ω −1 σc t+2 w 1 σc−1 t+1 n σn σc t+1 ¸−σl .
It is straightforward to see that the previous equation can be rewritten as follows:
α1 (qt+1nt+1)−σn = θ α2 · 1 − θ qt+1nt+1− Ω −1 σc t+2 w 1 σc−1 t+1 (qt+1nt+1) σn σc ¸−σl (13)
It follows that a decrease in child survival probability qt+1 is integrally compensated by a
proportional increase in fertility. Moreover, equations (11) and (12) giving optimal labor supply
and savings are independent of child mortality qt+1.¤
Some comments are in order. In our model, child mortality shocks induce a full insurance eect boosting fertility, while our speci cation does not allow them to generate any life-cycle type of mechanism. On the contrary, adult mortality induces a full life-cycle eect (driving labor
supply and savings down while pushing fertility upward), and no insurance behavior. In this sense, our model is able to generate a canonical picture where each type of mortality plays fully its natural role. Of course, this picture is likely to be altered if we depart from some essential assumptions made in our simple model. First of all, if children are allowed to work and to contribute to the (intertemporal) resources of the household and/or if they can support seniors via a social security system, then having children would also obey to intertemporal arbitrages. Second, even if we abstract away from the previous considerations, the picture can be quickly less canonical if we depart from a key assumption already signalled and extensively commented at the beginning of our presentation of the model, that is we assume that parents invest in their children after uncertainty about their survival is resolved This amounts to assuming a
speci c timing.10 If we depart from this assumption, then the picture is much more demanding
algebraically speaking, and more importantly, it will be de nitively less clear, opening the door to counter-factual behaviour. In particular, if we assume that parents invest in their children before uncertainty is resolved, then the optimal response of fertility to child mortality shocks
depends strongly on the value of the elasticity parameter σn. However, since the main aim of our
model is essentially to study the in uence of mortality shocks in a general equilibrium framework allowing for price eects to arise, we prefer to stick to this somewhat simpli ed and canonical picture, that turns out to be consistent with empirical evidence concerning child mortality impact on fertility, and will allow a much better identi cation of the main eects at work.
2.3 Impact of changing prices on optimal behavior
In this sub-section, before moving to general equilibrium we rst study how the optimal behavior depicted above is modi ed when prices exogenously move. As mentioned in the introduction, the literature on epidemics often outlines the role of wages in the propagation of the initial mortality shocks. Our intertemporal framework further has the virtue of exemplifying the role of another possibly important price, i.e. the interest rate.
First, a quick look at equations (10) and (11) is enough to identify a special case: when the
utility term with respect to consumption is logarithmic, that is when σc = 1, neither the wage
wt+1nor the interest rate Rt+2 matter in fertility and labor supply choices, and the price eects
are hence no longer active in this case. Indeed, both in equations (10) and (11), all the wage terms
disappear once we set σc= 1. In the case of the interest rate Rt+2, this result comes from the fact
that Rt+2 essentially operates through the term Ωt+2 = αθ1
µ 1 + pt+1α 1 σc 3 R 1 σc−1 t+2 ¶−σc , which is
trivially independent of Rt+2 under logarithmic preferences for consumption. The interpretation
of those results is straightforward. Higher wages traditionally induce 2 well-identi ed, competing eects: a positive income eect, increasing both consumption and leisure (and thus decreasing labor supply), and a substitution eect stemming from the increase in the opportunity cost of leisure, hence favorable to labor supply. These 2 opposite eects have the same magnitude when
σc = 1, and thus just oset each other in such a parametric case. Since labor supply is then
unaected, so is the fertility decision given the optimality condition (5). Similarly, when the
interest rate Rt+2 goes up, two eects emerge. Intertemporal substitution favors labor supply,
as before. At the same time, the relative price of future (senior) consumption with respect to
10In that way, it is important to notice that the insurance eect we describe in our model is somewhat dierent
from the one depicted in Kalemli-Ozcan (2003, 2008a), where it is the uncertain survival of adolescents that will trigger a precautionary demand for children, re ecting an insurance mechanism against more tardive deaths that cannot be easily replaced. As already signalled, we are closer to the mechanisms described in Kalemli-Ozcan (2008b).
present (young adult) consumption falls down as the interest rate rises, which goes against extra savings and labor supply. Again, both eects exactly compensate each other under logarithmic preferences for consumption.
Things are apparently much trickier once we allow for the wage eect and the interest rate
eect to be active, i.e. for σc6= 1. The next proposition summarizes the associated properties.
Proposition 4 For logarithmic preferences in consumption (σc = 1), the optimal fertility and
labor decisions are insensitive to wages and interest rate. For any σc positive and not equal
to 1, an increase in the wage wt+1 raises labor supply lt+1 and reduces fertility nt+1, if and only
σc< 1. A rise in the interest rate Rt+2 has the same properties.
Proof: Let us start with wages. Recall equation (13):
α1 (qt+1nt+1)−σn = θ α2 · 1 − θ qt+1nt+1− Ω −1 σc t+2 w 1 σc−1 t+1 (qt+1nt+1) σn σc ¸−σl .
While the left-hand side is unaected by wages, the right-hand side is. One can trivially see that the direction of the shift induced by an increase in wages is entirely determined by the position
of σc with respect to 1. If (and only if) σc < 1, an increment in wages increases the right-hand
side, which leads to a drop in optimal fertility to re-establish equation (13). The optimal labor supply response is slightly trickier to characterize, since both sides of equation (11) are aected by an increase in wages: α1 Ωt+2 w1−σc t+1 l−σt+1c = θ α2 · 1 − θΩσn1 t+2w σc−1 σn t+1 l σc σn t+1− lt+1 ¸−σl .
Suppose the wage wt+1 is rising and σc< 1. The left-hand side is shifted upwards, which induces
labor supply to increase to re-establish the equality (since, again, the left-hand side is decreasing
in lt+1). However, the right-hand side is also aected: it is actually shifted downwards, which
again induces a further increase in labor supply since the right-hand side is increasing in lt+1.
We get the opposite picture if σc> 1. As far as the interest rate is concerned, it can be readily
established that when Rt+2 increases, the term Ωt+2 = αθ1
µ 1 + pt+1α 1 σc 3 R 1 σc−1 t+2 ¶−σc goes in the
opposite direction if and only if σc < 1. To conclude about the impact of higher interest rates
on optimal fertility and labor supply, it is enough to observe that pt+1 and Rt+2 have the same
eect on Ωt+2 if and only if σc < 1. In the latter case, an increase in Rt+2 lowers Ωt+2, which
generates the same eects on fertility and labor supply as increasing pt+1. By Proposition 2,
and reversing its statement, we must have a decrease in fertility nt+1 and an increment in labor
supply.¤
The obtained results deserve a careful interpretation. Let us focus on the wage eect. An increase in wages induces a classic positive income eect, which tends to increase consumption (in both periods), leisure and the number of children. However, in our model an increase in the number of children and an increase in leisure are detrimental to each other : henceforth, the positive income eect has a non-trivial impact on fertility. On the other side, the other classic eect of higher wages is the substitution eect, increasing the opportunity cost of both leisure and rearing children. We thus get the typical opposition between income and substitution
eects à la Kremer and Chen (2002), which has in general an ambiguous eect on fertility.11
11It should be noted that Kremer and Chen (2002) directly choose a quasi-linear utility function for the
A key departure from Kremer and Chen (2002) is the intertemporal nature of our model. This characteristic is crucial, since it will readily enable us to understand the preeminent role of
the preference parameter σc. Indeed, in our story, individuals can take advantage of this wage
increase in t + 1 and only in t + 1: by increasing their labor supply to take advantage of higher wages in t + 1, they can transfer consumption to their old age in t + 2. Since the strength of
intertemporal substitution in consumption is measured by 1
σc, the lower σc, the more individuals
will be willing to transfer consumption to t + 2, the more they will work, and the lower leisure and fertility will be (via the needed reduction in the time devoted to rearing children). In our
model, the threshold value for σc is just one, that is logarithmic preferences in consumption
(in both ages): below this threshold, individuals work more and have less children, and above,
we have the opposite picture. We have a similar picture in the case of the interest rate. 12
2.4 Summary and relation to the fertility ambiguity under epidemic
shocks
Our simple model neatly illustrates why an epidemic shock has an ambiguous eect on fertility and labor supply. Indeed, if we interpret an epidemic shock as having direct consequences on survival probabilities and prices (for example on wages via the associated large cuts in labor supply), then our model shows that the total eect on fertility is a priori ambiguous when
σc < 1. On one side, a drop in both adult and children survival probabilities increases fertility,
adding to the well-known insurance eect put forward by Kalemli-Ozcan (2002) in the case of child mortality a life-cycle mechanism linking adult mortality to fertility decisions. On the other side, under certain parametric conditions, the increase in wages resulting from labor shortage has the exact opposite eect on fertility, i.e. decreasing it. If the interest rates also move, then a third eect has to be accounted for. As shown in Proposition 4, if the interest rates go up following an epidemic shock, then it will reinforce the wage eect described above, inducing a
larger drop in fertility.13
Whether the wage eect, neatly identi ed in our theory, can actually more than oset the mor-tality eect on fertility, as claimed by Young (2005) and questioned by Kalemli-Ozcan (2008a), is a crucial question that should deserve the maximal attention. However we have so far only studied optimal labor supply and fertility responses to exogenous shocks in prices. We shall now take a step further in the next section by closing the model and studying a general equilibrium version in which both wages and interest rates are endogenously determined. The main result of this section will outline the contrasting implications of child Vs adult mortality in general equilibrium, which will provide a kind of theoretical synthesis of Young's and Kalemli-Ozcan's approaches.
Finally, before moving to the characterization of these general equilibrium results, it is im-portant to recall that one could extend the analysis conducted on fertility (and labor supply) to education. If the typical trade-o quality vs quantity of children were introduced in the model, the wage eect of epidemics would have implied an increasing quality of children (that is, more education) as the number of children goes down. Hence, the increase in the opportunity cost
12Note that the simple results obtained (which themselves derive from the simplicity of the model) do not mean
that the other elasticity parameters are unimportant. One can for example notice that if σl, the elasticity of (marginal) utility of leisure with respect to the level of leisure, is increasingly large, the magnitude of the increase in labor supply will de nitely get lower (in the case σc < 1). We will however not comment further, since the focus of our model concerns the implication of price shocks on optimal fertility behavior.
13And it will be demonstrated in the next section that indeed, in the general equilibrium framework both wages
of rearing children, directly induced by the rise in wages, should increase educational attain-ment along with decreasing the fertility rate. The mortality eects yield the opposite outcomes. Again, the total eect on education depends on the strength of the price eects relative to the mortality eects.
3 Rational behavior under epidemics: accounting for
gen-eral equilibrium eects
In what follows, we place ourselves in the parametric case σc < 1, for which the wage eect
has been shown to operate in the direction opposite to the mortality eects in the previous section. Consistently with Zhang and Zhang (2005), we assume a production function of the Cobb-Douglas type:
Yt= Ktα (ltLt)1−α,
where Kt is the stock of capital available in period t, Lt the size of active population (assumed
homogenous) and α is the capital share. We shut down technological progress. We nally assume full capital depreciation in one period, such that at equilibrium, one should ensure that:
Kt+1= Ltst. (14)
Under perfect competition, the production factors are paid at their marginal productivities, which yields the two following price equations:
wt = (1 − α) µ Kt ltLt ¶α (15) and Rt= α µ Kt lt Lt ¶α−1 (16) To close the model, we nally need to observe that active population evolves according to the following law of motion:
Lt+1 = qtntLt. (17)
Indeed, people working in t + 1 are those who are young adults in t + 1, which therefore were children in period t. Abstracting away from child mortality, the number of children in period
t is equal to the number of surviving young adults (who are the Lt workers in that period)
times fertility of this generation, which is equal to nt. Accounting for child mortality gives the
demographic law of motion just above.
In our behavioral equations (11) to (13), the relevant prices for the generation born in t are
wt+1 and Rt+2. Using the equations (14) to (17) within our general equilibrium extension, one
obtains the following expressions for the two relevant prices:
wt+1 = (1 − α) µ Kt+1 lt+1 Lt+1 ¶α = (1 − α) µ st qtntlt+1 ¶α (18) and Rt+2 = α µ Kt+2 lt+2 Lt+2 ¶α−1 = α µ st+1 qt+1nt+1lt+2 ¶α−1 (19)
Equations (18) and (19) are enough to gather a very important result concerning child mor-tality.
Proposition 5 In general equilibrium, an increase in child mortality via a decrease in qt+1
has no impact on equilibrium prices, wt+1 and Rt+2. Therefore, a rise in child mortality does
unambiguously raise fertility in general equilibrium.
Indeed, a fall in qt+1 has not impact on qt+1 nt+1, lt+1 and st+1 by Proposition 3. From (18), it
follows that it has no impact on the wage wt+1. A priori, qt+1 may impact Rt+2 via the term
lt+2 in equation (19). Indeed, writing equation (11) one period ahead, one can see that lt+2 does
depend on wt+2 , which itself depends on qt+1 according to:
wt+2= (1 − α) µ Kt+2 lt+2Lt+2 ¶α = (1 − α) µ st+1 qt+1nt+1lt+2 ¶α
but again, here Proposition 3 is enough to conclude that such an eect is neutralized thanks to the
presence of the product qt+1 nt+1, which is independent of qt+1. So lt+2is unaected by changes in
the probability qt+1, which leaves the interest rate Rt+2also unaected. Therefore, chld mortality
has no general equilibrium price eect, and an increase in such a mortality unambiguously brings up fertility consistently with Kalemli-Ozcan (2003, 2008a).
Things are de nitely trickier for adult mortality, as re ected in the following proposition.
Proposition 6 In general equilibrium, an increase in adult mortality via a decrease in pt+1
cause wages wt+1 to unambiguously go up. Therefore, a rise in adult mortality has an ambiguous
eect on fertility in general equilibrium.
The possible opposite directions of the wage and adult mortality eects on fertility behavior had already been extensively commented in our partial equilibrium analysis, where positive shocks on prices were exogenously imposed. This property is here crucially shown to hold in general
equilibrium too. Indeed, by equation (18), one can immediately see that a drop in pt+1, causing
labor supply lt+1to decrease under xed prices (by Proposition 2), will further induce an increase
in wages. Such an increment in wages plays against fertility, and this second-round eect may counter-balance and possibly outweigh the direct positive eect of the adult survival probability drop on the same variable.
We deem the results stated in proposition 5 and 6 as highly relevant for the ongoing debate about the overall impact on fertility of epidemics such as the one of HIV/AIDS. Indeed, showing that the impact of a rise in child mortality on optimal fertility behavior is unambiguously positive through the absence of eect on prices (and particularly wages) at equilibrium (proposition 5) is indeed consistent with Kalemli-Ozcan (2002, 2008a), but also shows that the opposition between mortality eect and wage eect does not hold in the case of child mortality. It hence demonstrates that the initial debate between Kalemli-Ozcan and Young (Kalemli-Ozcan, 2008a) is somewhat awed, since the insurance eect channel had so far been argued to counteract the wage eect: we show that the two eects do not stem from the same type of mortality shock, since the rst one is associated to child mortality, while the second only arises in the case of adult mortality (proposition 6). In our view, disentangling and even isolating in such a way the two eects is a key result, once we consider that the mortality age pro les for major epidemics (Black Death,
Spanish u or AIDS) rather exhibit the preeminence of adult mortality.14 The debate should
14Recent demographic projections (see, for example, the UNAIDS (2004) annual report) are nevertheless
show-ing quite an alarmshow-ing trend for HIV-related child mortality in some Sub-Saharan African countries for the next two decades. However, it is undisputable that the vast majority of AIDS-related deaths are active adults.
then rather focus on the opposing positive direct eect of adult mortality on fertility behavior
(through the previously identi ed life-cycle channel15) and the negative, second-round eect
triggered by the resulting endogenous increase in wages. Our model shows that if the wage eect associated to this kind of mortality shock is strong enough, then the optimal fertility decision might end up being reduced, and not augmented by the epidemics.
One should also notice this last possibility is even more likely if the interest rate eect ends up reinforcing the wage eect. From equation (19), one can see that the interest rate is also
likely to go up after the adult epidemic shock because a drop in pt+1 lowers savings, st+1 and
increases fertility, nt+1 by Proposition 2: Both moves increase the interest rate Rt+2, which by
Proposition 4, reinforces the above outlined wage eect.
Unfortunately, it is not possible to bring out analytical results characterizing nely when the wage eect does dominate the direct mortality eect on fertility, neither in the short run nor in the stationary equilibrium. The model is certainly solvable in general equilibrium under logarithmic preferences, but as already outlined before this case is uninteresting as far as the question at hand is concerned, since it neutralizes the prices eects on optimal decisions (as properly shown in Proposition 4). Rather than looking at rough calibrations of the model, we now complement our theoretical analysis with an empirical analysis, applied to the HIV/AIDS Sub-Saharan case. We will hence see that in this particular epidemic episode, the negative wage eect on fertility indeed seems to dominate the positive life-cycle eect in the case of an increase in adult mortality.
4 Empirical analysis
Our empirical speci cation will focus on dierentiating the two possible dierent mortality chan-nels (child and adult mortality) that can arise in an epidemic shock, and on exemplifying their dierent overall impact on fertility behavior. This precise identi cation of the role of the two dierent mortality shocks will enable us to nally tackle the overall impact of HIV, that has been found to in uence the fertility behavior through both the child and adult mortality channels.
4.1 Empirical model and data
In addition to mortality, fertility should be in uenced by education, real income per capita and con ict occurrence. A higher average number of schooling years is likely to increase the opportunity cost of childbearing for women, hence education is expected to have a negative impact on the fertility rate. Similarly, the net costs of childrearing, e.g. housing, nursing and training costs, tend to be higher in more developed countries displaying a higher level of real income per capita (Becker, 1992). Last, con ict occurrence may decrease the willingness or the feasibility of having children since current and future upbringing conditions may not be perceived as optimal (Agadjanian and Prata, 2001). The following model will therefore be estimated:
Total fertility ratet
i = β1Ln(Child mortality)ti+ β2Ln(Adult mortality)ti
+β3Ln(Real GDP per capita)ti+ β4Educationti + β5Con ict occurrenceti
+Tt+ ²t
i (1)
where Tt are country-invariant time-speci c t xed eects and ²t
i = Ci+ υit, with time-invariant
country-speci c i xed eects Ci and idiosyncratic shocks υit. As previously stated and in line
with our theoretical model, fertility is expected to be positively in uenced by child mortality and negatively in uenced by income per capita, education and the occurrence of a territorial con ict. The impact of adult mortality on the other hand is ambiguous.
The total fertility rate is the number of children that a woman would have if she lived through all of her child-bearing years and experienced the current age-speci c fertility rates at each age. Data come from the United Nations Population Division (2007). Child mortality (rate per 1000 live births) is de ned as the probability of dying before age 5, if subject to current age-speci c mortality rates. Data come from the World Health Organization. Adult mortality rate is the probability of dying between the ages of 15 and 60 years (per 1000 population) per year, if subject to the age-speci c mortality rates of the reporting year. Data on male and female adult mortality rates come from the World Bank (2007) and have been combined into an average mortality rate by weighing each mortality rate by the gender share in the active population. Real GDP per capita and education, the latter measured by average schooling years in population aged 25 or over, come from Baier et al. (2006). Finally, the armed con icts dataset developed by Uppsala University (Eriksson and Wallensteen, 2004) provides a measure of the intensity and length of a territorial con ict. The con ict occurrence variable takes the value of 1 if a minor armed con ict occurs, 2 if an intermediate armed con ict occurs, 3 if a war occurs, on the country's territory. For ease of interpretation the variable has been re-scaled from range 0 to 1.
Data on the total fertility rate are reported as ve year averages by the UNPD. Hence, a ve-year period panel covering the apparition and spread of the HIV epidemic in Sub-Saharan Africa is constructed. The quinquennial periods are 1980-1984, 1985-1989, 1990-1994, 1995-1999, 2000-2004. However, besides the data on con ict occurrence, which are available for consecutive years and have been averaged over each ve-year period, values of other variables are only available for the years 1980, 1990 and 2000. Hence they have been interpolated in order to obtain beginning of period values for 1985 and 1995. In Appendix 1, it is shown that the results of this paper hold when using non-interpolated ten-year period panel data from 1960 to 2004.
Concerns about the reliability of the data may be raised. The data generated by Baier et al. (2006) have been rarely used in the literature and mortality rates, especially those of the adult population, are certainly measured with substantial error. Therefore, as commonly done in the literature on the growth impact of education (see for instance Cohen and Soto (2007)), reliability indicators measuring the signal-to-total variance ratio of a variable have been constructed. They can be found in Appendix 2. They show that the informational content of the data series used in this paper is very high, even when expressed in changes, and that the spatio-temporal coverage of other data sources is much narrower.
The only two channels identi ed by our model through which the disease should have an eect on fertility are the child mortality channel and the adult mortality channel. A simple way of testing such a hypothesis is to directly include a measure of HIV prevalence in equation 1. An absence of signi cance of its coe cient will be consistent with a model in which the eect of HIV is completely transmitted through the mortality variables. On the other hand, signi cance will imply that HIV prevalence exerts a direct eect on fertility, beyond its impact on mortality. Time-series data on HIV prevalence in adult population, the percentage of people aged 15-49 who are infected with HIV, have been obtained from Karen Stanecki, UNAIDS senior epidemiologist. The time series have been estimated in November 2006 with the use of the UNAIDS Estimation
and Projection Package (EPP) 2005, which ts an epidemic model to all available estimates of HIV prevalence in order to produce a country-speci c epidemic curve that describes the evolution
of adult HIV prevalence rates over the 1980-2005 period.16 Since the time-series for each country
have been generated at the same time they are not plagued by temporal inconsistency caused by changes in assumptions, methodologies and data used. However, one major limitation of the EPP is related to the non-representative nature of the data used since estimates of HIV prevalence in adult population are mainly derived from data on the proportion of seropositive females among pregnant women attending urban or peri-urban antenatal clinics. Potential biases associated with the use of antenatal clinic data include the selection for sexual activity and absence of contraceptive use, the lower fertility of HIV-infected women, the joint determination of antenatal clinic attendance and HIV status, the under-representation of smaller rural sites in surveillance systems and how well prevalence levels among pregnant women represent those among men (Walker et al., 2004). Where countries' surveillance systems do not cover adequately rural areas, in which HIV prevalence is expected to be lower than in urban areas, the non-urban prevalence produced by the EPP is adjusted downwards by 20% to re ect this bias. Adjustments for expansion of surveillance systems into lower prevalence areas and for turnover in concentrated epidemics are also implemented. Finally and most importantly, when general population survey data are available, trends tted from antenatal data have been recalibrated to adjust urban and rural HIV levels to those measured in the population-based survey. In appendix 1, it is shown that these estimates of HIV prevalence appear reliable, certainly because the calibration of the data to national population surveys had occurred for most of the countries included in the data
used in the reliability analysis.17 Hence, the robustness of the results will be assessed by either
interacting the HIV variable with a dummy which takes the value of one when the prevalence rate has been adjusted, or by restricting the sample to the countries for which calibration has occurred.
Data are available for 39 SSA countries over the 1980-2004 period. Summary statistics are given in tables 1 and 2. Several regressions diagnostics have been carried out. According to a Cook's D test, in uential outliers have been removed from the sample, Ramsey (1969)'s RESET or Pregibon (1980)'s Link tests suggests that the model is well speci ed and the low value of the mean variance in ation factor indicates that multicollinearity is not a problem. However, most variables in equation 1 are likely to be endogenous to the fertility rate and undoubtedly measured
with error.18 To correct both biases in the context of panel data models, when no obvious external
instruments are available, internal instruments based on the lags of the instrumented variables can be used. This can be done through the use of the system-GMM estimator (Arellano and Bover, 1995; Blundell and Bond, 1998). Equation 1 will be simultaneously estimated in rst dierence and level and to ensure consistent estimates of the coe cients, the equation in rst-dierence (level) will be instrumented by the lagged levels ( rst- rst-dierences) of the variables.
16See Ghys et al. (2004) and Brown et al. (2006) for a clear presentation of the EPP. The approach is dierent
for low-level or concentrated epidemics in which HIV is concentrated in groups with high-risk behaviours. For these epidemics, where the HIV prevalence rate is considered to be below 1% in pregnant women in urban areas, because transmission is assumed to occur mainly in groups at high risk of HIV infection, estimates for populations who are most exposed to HIV/AIDS are combined to produce an overall estimate of adult prevalence (Lyerla et al., 2006).
17Nevertheless, note that Grassly et al. (2004) show that estimates based on antenatal sentinel surveillance
provide a good approximation of HIV prevalence in adults in the local community and Young (2007), using antenatal and community data for 50 regions in 8 African countries, does not nd a signi cant dierence between community and antenatal infection rates.
18Note that measurement errors constant over time but speci c to each country will be absorbed into the
Variable Mean Std. Dev. Min. Max. N
Fertility 5.99 1.16 1.91 8.10 185
Average years of schooling 3.02 1.87 0.33 8.60 185
Child Mortalityln 5.10 0.46 3.00 5.82 185
Adult Mortalityln 6.03 0.22 5.16 6.60 185
Real GDP per capitaln 6.81 0.65 5.74 8.83 185
Territorial con ict 0.18 0.31 0.00 1.00 185
HIV prevalenceln 1.00 0.95 0.00 3.39 185
HIV prevalence adjustedln 1.11 0.96 0.00 3.39 95
Notes: ln: variable in logarithms. `1' has been added to the HIV prevalence rate in order to deal with zero values (15% of observations in 1980) .
Table 1: Summary statistics
Under the assumption that these lagged values are not correlated with the error term, they will be appropriate instruments. The validity of the instruments will be tested through Hansen (1982) tests of over-identifying restrictions and an Arellano and Bond (1991) test of serial correlation of the dierenced error term. In order not to over t the instrumented variables, which may bias the results towards those of an uninstrumented regression and weaken the Hansen tests, the GMM instruments are restricted to period t − 2 (t − 1) lagged levels ( rst-dierences) of each variable
for the rst-dierence (level) equation.19 In addition, robustness of the results to the number
of instruments will be investigated by collapsing the number of instruments, i.e. the instrument set will only contain two instruments per instrumented variable instead of including all available valid lags (see Roodman (2007)).
19Roodman (2004) indicates as a rule of thumb that the number of instruments should not exceed the number
Child M. Child M. Child M. A dult M. A dult M. A dult M. ∆ Child M. ∆ A dult M. HIV Prev alence (%) 1980 1990 2000 1980 1990 2000 1990-2000 (%) 1990-2000 (%) 2000 (%) Coun try Botsw ana 84 58 101 309 411 678 74 65 28 Lesotho 155 120 91 322 438 684 -24 56 24* Zim bab w e 108 80 117 355 287 733 46 155 23* South Africa 91 60 63 432 397 496 5 25 18* Namibia 108 86 69 396 345 493 -20 43 17 Zam bia 155 180 182 447 405 692 1 71 17* Mala wi 265 241 188 387 457 614 -22 34 14 Mozam bique 220 235 178 413 367 534 -24 45 14 Cen tral African Republic 189 180 180 480 431 618 0 43 11 Gab on 115 92 91 430 367 375 -1 2 8 Ken ya 115 97 120 378 322 523 24 62 7* Uganda 185 160 145 429 493 562 -9 14 7* T anzania 175 163 165 410 408 492 1 20 7* Camero on 173 139 166 451 395 472 19 19 6* Congo 125 110 108 352 321 465 -2 45 6* Côte d'Iv oire 172 157 188 385 324 454 20 40 5 Rw anda 219 173 203 455 450 543 17 21 4* Angola 265 260 260 512 466 484 0 4 4 Guinea-bissau 290 253 215 526 539 426 -15 -21 4* Nigeria 216 235 205 494 439 481 -13 10 4 Lib eria 235 235 235 226 378 495 0 31 3 Burundi 195 190 190 443 418 526 0 26 3* Chad 225 203 200 501 441 471 -1 7 3 T ogo 175 152 142 415 355 329 -7 -7 3 Ghana 157 125 100 367 302 334 -20 10 2* Benin 214 185 160 440 408 300 -14 -26 2 Burkina faso 247 210 207 415 384 426 -1 11 2* Ethiopia 220 204 176 446 403 429 -14 6 2* Gam bia 231 154 128 524 480 313 -17 -35 2 Mali 300 250 224 408 392 344 -10 -12 2* Sierra leone 336 302 286 533 546 411 -5 -25 2* Sudan 142 120 97 500 431 308 -19 -28 2 Guinea 300 240 175 549 512 298 -27 -42 2* Niger 320 320 270 508 465 354 -16 -24 1* Senegal 218 148 139 550 443 298 -6 -33 1* Somalia 225 225 225 456 427 392 0 -8 1 Mauritania 175 183 183 460 402 331 0 -18 1 Madagascar 175 168 137 315 405 310 -18 -23 0 Mauritius 40 25 20 214 187 173 -20 -7 0 A verage 194 172 162 427 406 453 -3 14 7 Notes: M: Mortalit y. * indicates that HIV prev alence rates ha ve been calibrated to national population surv eys. Table 2: Coun tries in the sample
4.1.1 Results
Results are presented in table 3. The validity of the instruments is never rejected, all control
variables have the expected sign and most of them are signi cant.20 In line with our theoretical
model, child mortality exerts a positive impact on fertility whereas we empirically nd that the opposite is true for adult mortality (column 1). Columns 2 and 3 show that the impacts of male adult mortality rate and female adult mortality on fertility are about the same. Finally, in column 4, the sensitivity of results and speci cation tests to a reduction in the number of instruments is investigated by collapsing the instruments. Although the e ciency of the regression estimates decreases, results are fairly unaected by this robustness check, at the exception of the territorial con ict variable, whose negative eect becomes highly signi cant.
Kalemli-Ozcan (2008a) nds that the logarithm of the HIV prevalence exerts a direct positive eect on fertility, beyond the HIV-related rise in infant mortality. This issue is investigated in table 4. Keeping the instruments collapsed, the logarithm of the HIV prevalence rate is included
among other determinants of fertility in column 5.21 Its coe cient is indeed positive but small
and highly insigni cant. However the reliability of HIV prevalence rates not calibrated to national population surveys is questionable. Robustness of these results are therefore examined by either interacting the HIV variable with a dummy which takes the value of one when the prevalence rate has been adjusted (column 6) or by restricting the sample to the countries for which calibration has occurred (column 7). In both cases, the null of a direct impact of HIV prevalence on fertility
cannot be rejected.22 Column 7 shows that even with half the initial sample, variables of interest
remain signi cant with the expected sign and it cannot be rejected that their coe cients equal the estimates obtained in column (1). The absence of a direct impact of HIV prevalence suggests that the mortality channels capture reasonably well the total eect of this disease on fertility.
In order to estimate the full impact of a rise in HIV prevalence on fertility through both child and adult mortality channels, simple nite distributed lag models are estimated. It makes sense to allow for lagged eects of HIV prevalence on mortality since full progression from infection to AIDS death in the absence of competing causes of mortality takes at maximum about 10 years in the case of children, which are not necessarily born at the time of the maternal infection, and 15-20 years in the case of adults (Stover, 2004). As a benchmark, a nite distributed lag model of order 3 is rst estimated and afterwards, the appropriate number of lags is chosen according
20Other variables, included in Conley et al. (2006) or Kalemli-Ozcan (2008a) have been tested: population
den-sity, urbanisation rate, area-weighted Green Revolution modern variety crops. None was found to be signi cant.
21`1' has been added to the HIV prevalence rate in order to deal with zero values (15% of observations for the
period 1980-1984).
22As another robustness check, the instrumental variable strategy of Kalemli-Ozcan (2008a) was adopted. We
ran between regressions, with the logarithm of the HIV prevalence instrumented by the distance from the capital city of each country to the capital city of the Democratic Republic of Congo, alleged birthplace of the epidemic. The coe cient of the HIV prevalence rate remained insigni cant and the validity of the instrument was rejected by the Cragg-Donald F statistic. The positive impact found by Kalemli-Ozcan (2008a) may result from the fact that distance from the Democratic Republic of Congo is positively correlated with the share of muslims in total population in 1980 (r=0.54). Thanks to circumcision, muslims may have lower HIV prevalence than non-Muslims and their religious beliefs may positively aect their fertility behaviour. As pointed out by Murray (2006), an instrumental variable strategy is no more valid if a variable correlated with either the endogenous variable or the instrument has been omitted from the econometric model. In rst-stage regressions, the share of muslims in total population entered signi cantly with a negative sign and once this variable was included, the bilateral distance was found to be not signi cant any more, suggesting indeed an omitted variable bias. It should further be noted that even if a good instrument could be found for HIV prevalence, estimators will still be biased and inconsistent if other determinants of fertility are endogenous.